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Theorem seqcaopr3 12776
Description: Lemma for seqcaopr2 12777. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
seqcaopr3.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqcaopr3.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seqcaopr3.3 (𝜑𝑁 ∈ (ℤ𝑀))
seqcaopr3.4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
seqcaopr3.5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)
seqcaopr3.6 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
seqcaopr3.7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
Assertion
Ref Expression
seqcaopr3 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Distinct variable groups:   𝑘,𝑛,𝑥,𝑦,𝐹   𝑘,𝐻,𝑛   𝑘,𝑁,𝑛,𝑥,𝑦   𝜑,𝑘,𝑛,𝑥,𝑦   𝑘,𝐺,𝑛,𝑥,𝑦   𝑘,𝑀,𝑛,𝑥,𝑦   𝑄,𝑘,𝑛,𝑥,𝑦   + ,𝑛,𝑥,𝑦   𝑆,𝑘,𝑥,𝑦
Allowed substitution hints:   + (𝑘)   𝑆(𝑛)   𝐻(𝑥,𝑦)

Proof of Theorem seqcaopr3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 seqcaopr3.3 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 12291 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6148 . . . . 5 (𝑧 = 𝑀 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑀))
5 fveq2 6148 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑀))
6 fveq2 6148 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑀))
75, 6oveq12d 6622 . . . . 5 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
84, 7eqeq12d 2636 . . . 4 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
98imbi2d 330 . . 3 (𝑧 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))))
10 fveq2 6148 . . . . 5 (𝑧 = 𝑛 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑛))
11 fveq2 6148 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑛))
12 fveq2 6148 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑛))
1311, 12oveq12d 6622 . . . . 5 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
1410, 13eqeq12d 2636 . . . 4 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))))
1514imbi2d 330 . . 3 (𝑧 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))))
16 fveq2 6148 . . . . 5 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘(𝑛 + 1)))
17 fveq2 6148 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18 fveq2 6148 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1917, 18oveq12d 6622 . . . . 5 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))
2016, 19eqeq12d 2636 . . . 4 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1)))))
2120imbi2d 330 . . 3 (𝑧 = (𝑛 + 1) → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
22 fveq2 6148 . . . . 5 (𝑧 = 𝑁 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑁))
23 fveq2 6148 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24 fveq2 6148 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
2523, 24oveq12d 6622 . . . . 5 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
2622, 25eqeq12d 2636 . . . 4 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
2726imbi2d 330 . . 3 (𝑧 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))))
28 eluzfz1 12290 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
291, 28syl 17 . . . . . 6 (𝜑𝑀 ∈ (𝑀...𝑁))
30 seqcaopr3.6 . . . . . . 7 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
3130ralrimiva 2960 . . . . . 6 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
32 fveq2 6148 . . . . . . . 8 (𝑘 = 𝑀 → (𝐻𝑘) = (𝐻𝑀))
33 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
34 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
3533, 34oveq12d 6622 . . . . . . . 8 (𝑘 = 𝑀 → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
3632, 35eqeq12d 2636 . . . . . . 7 (𝑘 = 𝑀 → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
3736rspcv 3291 . . . . . 6 (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
3829, 31, 37sylc 65 . . . . 5 (𝜑 → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀)))
39 eluzel2 11636 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
401, 39syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
41 seq1 12754 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻𝑀))
4240, 41syl 17 . . . . 5 (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻𝑀))
43 seq1 12754 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
44 seq1 12754 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺𝑀))
4543, 44oveq12d 6622 . . . . . 6 (𝑀 ∈ ℤ → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
4640, 45syl 17 . . . . 5 (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
4738, 42, 463eqtr4d 2665 . . . 4 (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
4847a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
49 oveq1 6611 . . . . . 6 ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
50 elfzouz 12415 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
5150adantl 482 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
52 seqp1 12756 . . . . . . . 8 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))))
5351, 52syl 17 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))))
54 seqcaopr3.7 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
55 fzofzp1 12506 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
5655adantl 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
5731adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
58 fveq2 6148 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐻𝑘) = (𝐻‘(𝑛 + 1)))
59 fveq2 6148 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
60 fveq2 6148 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
6159, 60oveq12d 6622 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6258, 61eqeq12d 2636 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
6362rspcv 3291 . . . . . . . . . 10 ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
6456, 57, 63sylc 65 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6564oveq2d 6620 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
66 seqp1 12756 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
67 seqp1 12756 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
6866, 67oveq12d 6622 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑀) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
6951, 68syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7054, 65, 693eqtr4rd 2666 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
7153, 70eqeq12d 2636 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) ↔ ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1)))))
7249, 71syl5ibr 236 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1)))))
7372expcom 451 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
7473a2d 29 . . 3 (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))) → (𝜑 → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
759, 15, 21, 27, 48, 74fzind2 12526 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
763, 75mpcom 38 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  cfv 5847  (class class class)co 6604  1c1 9881   + caddc 9883  cz 11321  cuz 11631  ...cfz 12268  ..^cfzo 12406  seqcseq 12741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742
This theorem is referenced by:  seqcaopr2  12777  gsumzaddlem  18242
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